Manifolds and Differential Geometry

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Jeffrey M. Lee

Differential geometry began as the study of curves and surfaces using
the methods of calculus. In time, the notions of curve and surface were
generalized along with associated notions such as length, volume, and
curvature. At the same time the topic has become closely allied with
developments in topology. The basic object is a smooth manifold, to
which some extra structure has been attached, such as a Riemannian
metric, a symplectic form, a distinguished group of symmetries, or a
connection on the tangent bundle.

This book is a graduate-level introduction to the tools and structures
of modern differential geometry. Included are the topics usually found
in a course on differentiable manifolds, such as vector bundles,
tensors, differential forms, de Rham cohomology, the Frobenius theorem
and basic Lie group theory. The book also contains material on the
general theory of connections on vector bundles and an in-depth chapter
on semi-Riemannian geometry that covers basic material about Riemannian
manifolds and Lorentz manifolds. An unusual feature of the book is the
inclusion of an early chapter on the differential geometry of
hypersurfaces in Euclidean space. There is also a section that derives
the exterior calculus version of Maxwell's equations.

The first chapters of the book are suitable for a one-semester course on
manifolds. There is more than enough material for a year-long course on
manifolds and geometry.

Readership

Graduate students and research mathematicians interested in
differential geometry.

Reviews & Endorsements

This book is
certainly a welcome addition to the literature. As noted, the author has an
on-line supplement, so the interested reader can follow up on the development
of further topics and corrections. One cannot begin to imagine the Herculean
amount of work that went into producing a volume of this size and scope, over
660 pages! Future generations will be in the author's debt.

Differential geometry began as the study of curves and surfaces using
the methods of calculus. In time, the notions of curve and surface were
generalized along with associated notions such as length, volume, and
curvature. At the same time the topic has become closely allied with
developments in topology. The basic object is a smooth manifold, to
which some extra structure has been attached, such as a Riemannian
metric, a symplectic form, a distinguished group of symmetries, or a
connection on the tangent bundle.

This book is a graduate-level introduction to the tools and structures
of modern differential geometry. Included are the topics usually found
in a course on differentiable manifolds, such as vector bundles,
tensors, differential forms, de Rham cohomology, the Frobenius theorem
and basic Lie group theory. The book also contains material on the
general theory of connections on vector bundles and an in-depth chapter
on semi-Riemannian geometry that covers basic material about Riemannian
manifolds and Lorentz manifolds. An unusual feature of the book is the
inclusion of an early chapter on the differential geometry of
hypersurfaces in Euclidean space. There is also a section that derives
the exterior calculus version of Maxwell's equations.

The first chapters of the book are suitable for a one-semester course on
manifolds. There is more than enough material for a year-long course on
manifolds and geometry.

Graduate students and research mathematicians interested in
differential geometry.

Reviews:

This book is
certainly a welcome addition to the literature. As noted, the author has an
on-line supplement, so the interested reader can follow up on the development
of further topics and corrections. One cannot begin to imagine the Herculean
amount of work that went into producing a volume of this size and scope, over
660 pages! Future generations will be in the author's debt.